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Popular >

integral of x^4(1+x^5)^8

Solution

\int x^{4} (1 + x^{5})^{8} d x

Solution

\frac{x ^{5}}{5} + \frac{4 x ^{10}}{5} + \frac{28 x ^{15}}{15} + \frac{14 x ^{20}}{5} + \frac{14 x ^{25}}{5} + \frac{28 x ^{30}}{15} + \frac{4 x ^{35}}{5} + \frac{x ^{40}}{5} + \frac{x ^{45}}{45} + C

Solution steps

\int x^{4} (1 + x^{5})^{8} d x

Expand

x^{4} (1 + x^{5})^{8} : x^{4} + 8 x^{9} + 28 x^{14} + 56 x^{19} + 70 x^{24} + 56 x^{29} + 28 x^{34} + 8 x^{39} + x^{44}

= \int x^{4} + 8 x^{9} + 28 x^{14} + 56 x^{19} + 70 x^{24} + 56 x^{29} + 28 x^{34} + 8 x^{39} + x^{44} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{4} d x + \int 8 x^{9} d x + \int 28 x^{14} d x + \int 56 x^{19} d x + \int 70 x^{24} d x + \int 56 x^{29} d x + \int 28 x^{34} d x + \int 8 x^{39} d x + \int x^{44} d x

\int x^{4} d x = \frac{x ^{5}}{5}

\int 8 x^{9} d x = \frac{4 x ^{10}}{5}

\int 28 x^{14} d x = \frac{28 x ^{15}}{15}

\int 56 x^{19} d x = \frac{14 x ^{20}}{5}

\int 70 x^{24} d x = \frac{14 x ^{25}}{5}

\int 56 x^{29} d x = \frac{28 x ^{30}}{15}

\int 28 x^{34} d x = \frac{4 x ^{35}}{5}

\int 8 x^{39} d x = \frac{x ^{40}}{5}

\int x^{44} d x = \frac{x ^{45}}{45}

= \frac{x ^{5}}{5} + \frac{4 x ^{10}}{5} + \frac{28 x ^{15}}{15} + \frac{14 x ^{20}}{5} + \frac{14 x ^{25}}{5} + \frac{28 x ^{30}}{15} + \frac{4 x ^{35}}{5} + \frac{x ^{40}}{5} + \frac{x ^{45}}{45}

Add a constant to the solution

= \frac{x ^{5}}{5} + \frac{4 x ^{10}}{5} + \frac{28 x ^{15}}{15} + \frac{14 x ^{20}}{5} + \frac{14 x ^{25}}{5} + \frac{28 x ^{30}}{15} + \frac{4 x ^{35}}{5} + \frac{x ^{40}}{5} + \frac{x ^{45}}{45} + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x^4(1+x^5)^8 ?
The integral of x^4(1+x^5)^8 is